Abstract
The pure $R^2$ gravity is equivalent to Einstein gravity with cosmological constant and a massless scalar field and it further possesses the so-called restricted Weyl symmetry which is a symmetry larger than scale symmetry. To incorporate matter, we consider a restricted Weyl invariant action composed of pure $R^2$ gravity, SU(2) Yang-Mills fields and a non-minimally coupled massless Higgs field (a triplet of scalars). When the restricted Weyl symmetry is spontaneously broken, it is equivalent to an Einstein-Yang-Mills-Higgs (EYMH) action with a cosmological constant and a massive Higgs non-minimally coupled to gravity i.e. via a term $\tilde{\xi} R |\Phi|^2$. When the restricted Weyl symmetry is not spontaneously broken, linearization about Minkowski space-time does not yield gravitons in the original $R^2$ gravity and hence it does not gravitate. However, we show that in the broken gauge sector of our theory, where the Higgs field acquires a non-zero vacuum expectation value, Minkowski space-time is a viable gravitating background solution. We then obtain numerically gravitating magnetic monopole solutions for non-zero coupling constant $\tilde{\xi}=1/6$ in three different backgrounds: Minkowski, anti-de Sitter (AdS) and de Sitter (dS), all of which are realized in our restricted Weyl invariant theory.
Highlights
In the last few years there has been an interest in studying pure R2 gravity
The pure R2 gravity is equivalent to Einstein gravity with cosmological constant and a massless scalar field, and it further possesses the so-called restricted Weyl symmetry, which is a symmetry larger than scale symmetry
When the restricted Weyl symmetry is spontaneously broken, it is equivalent to an Einstein-Yang-MillsHiggs (EYMH) action with a cosmological constant and a massive Higgs nonminimally coupled to gravity, i.e., via a term ξRjΦ⃗ j2
Summary
In the last few years there has been an interest in studying pure R2 gravity (i.e., gravity action solely made out of R2 with no additional R term or cosmological constant, where R is the Ricci scalar). The extra terms include a cosmological constant, a nonminimal coupling term RjΦ⃗ j2, a massless scalar field φ, and an interaction term between φ and the Higgs field This action yields gravitating magnetic monopole (and black hole) solutions in dS, AdS, and Minkowski backgrounds. The above action is restricted Weyl invariant [3,5]; i.e., it is invariant under the transformation gμν → Ω2gμν; Φ⃗ → Φ⃗ =Ω; Aiμ → Aiμ with □Ω 1⁄4 0; ð2Þ where the conformal factor ΩðxÞ is a real smooth function This symmetry forbids a mass term for the Higgs, an Einstein-Hilbert term, as well as a cosmological constant term in (1). On how it couples to the Higgs field; it determines the nature of the interaction term between φ (or ψ) and Φ⃗ in (6)
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